Is this equation of hyperbolic type?

Question

I want to now whether this equation is of hyperbolic type: $$(1-\partial_{xx})y_{tt}+y_{xxxx}=0$$ with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$ I would say that the answer is yes. By taking the principal symbols we get $$-\tau^2+\frac{\xi^4}{1+\xi^2}=0$$ which has the same behavior as $$-\tau^2+\xi^2=0$$ which is clearly the wave equations for high frequency. Am I right ? Does the boundary conditions affects the type of PDEs ? Thank you.

Answer 1

You made a confusion between the symbol (here $-(1+\xi^2)\tau^2+\xi^4$), and the principal symbol, which gathers the monomials of highest degree. Since the latter is $\xi^4-\xi^2\tau^2$, which splits into linear factors over the reals, your equation is hyperbolic.

However, "being hyperbolic" is never a complete sentence in the theory. One must say instead "being hyperbolic in the direction of some (future) cone". In $1+1$ dimension, this is any convex cone that is transverse to the characteristic lines. Then you can impose a Cauchy data along any non-characteristic line. Unfortunately $t=0$ is characteristic for your PDE, because of the linear factor $\xi$ in the ppal symbol.

What you should do: rewrite the PDE as an abstract second-order differential equation $y_{tt}=Ay$ with $A=(\partial_x^2-1)^{-1}\partial_x^4$ associated with the boundary condition. Look at the spectrum of $A$. Prove that its real part is bounded from above, and perhaps that it is self-adjoint. Then apply some abstract theorem about continuous semi-groups.